Lemma from Auerbach

The Auerbach Lemma (after Herman Auerbach ) is a statement of functional analysis . It says that there is always an Auerbach basis in an n -dimensional normed vector space . The set in E is called an Auerbach basis of E if exist in the dual space of E with norm 1, so that for all . It is the Kronecker delta if so equal to 1, and is 0 otherwise. ${\displaystyle (E,\|\cdot \|)}$ ${\displaystyle e_{1},\ldots ,e_{n}}$ ${\displaystyle f_{1},\ldots ,f_{n}}$ ${\displaystyle \!\ E'}$ ${\displaystyle f_{j}(e_{k})=\delta _{j,k}}$ ${\displaystyle 1\leq j,k\leq n}$ ${\displaystyle \delta _{j,k}}$ ${\displaystyle j=k}$

Because of the equations , the vectors are linearly independent , so they form a basis of the vector space. The proof uses tools from linear algebra and elementary analysis. ${\displaystyle f_{j}(e_{k})=\delta _{j,k}}$ ${\displaystyle e_{j}}$

In the case of the Euclidean norm on a finite-dimensional vector space or the unit vectors satisfy the statement of the lemma. In addition, Auerbach's lemma makes a statement about any vector space norm and is then not as obvious as the case of the Euclidean vector space. ${\displaystyle {\mathbb {R} }^{n}}$ ${\displaystyle {\mathbb {C} }^{n}}$ ${\displaystyle (1,0,\ldots ,0),\ldots ,(0,\ldots ,0,1)}$

In Hilbert spaces , every orthonormal basis is an Auerbach basis . The functionals are used as in the above lemma . In some situations, including the following application, an Auerbach basis can act as a substitute for orthonormal bases. ${\displaystyle (e_{i})_{i}}$ ${\displaystyle f_{j}}$ ${\displaystyle \langle \,\cdot \,,e_{j}\rangle }$

application

The following statement about not necessarily finite-dimensional spaces shows how this lemma can be used.

If E is a normalized space and F is an n -dimensional subspace , then there is a continuous projection P from E onto F with . ${\displaystyle \|P\|\leq n}$

According to the lemma, the n -dimensional subspace F has an Auerbach basis with and according to Hahn-Banach's theorem there is with and . By recalculating it can then be shown that ${\displaystyle e_{1},\ldots ,e_{n}}$ ${\displaystyle f_{1},\ldots ,f_{n}\in F'}$ ${\displaystyle g_{j}\in E'}$ ${\displaystyle g_{j}|_{F}=f_{j}}$ ${\displaystyle \|g_{j}\|=1}$

${\displaystyle P:x\mapsto \sum _{j=1}^{n}g_{j}(x)e_{j}}$

is a projection from E onto F with . ${\displaystyle \|P\|\leq n}$

This theorem can be improved considerably; according to Kadets-Snobar's theorem , there are even projections with norm less than or equal , but the proof of this statement is much more difficult. ${\displaystyle {\sqrt {n}}}$

literature

• Reinhold Meise, Dietmar Vogt: Introduction to Functional Analysis , Vieweg, Braunschweig 1992, ISBN 3-528-07262-8 .